U.S. patent application number 10/704105 was filed with the patent office on 2005-05-12 for dynamically balanced walk behind trowel.
This patent application is currently assigned to Wacker Corporation. Invention is credited to Dauffenbach, Darrin W., Goldberg, Richard D., Kruepke, Gregory, Lutz, Todd J..
Application Number | 20050100404 10/704105 |
Document ID | / |
Family ID | 34435587 |
Filed Date | 2005-05-12 |
United States Patent
Application |
20050100404 |
Kind Code |
A1 |
Lutz, Todd J. ; et
al. |
May 12, 2005 |
Dynamically balanced walk behind trowel
Abstract
A walk behind rotary trowel is configured to be "dynamically
balanced" so as to minimize the forces/torque that the operator
must endure to control and guide the trowel. Characteristics that
are accounted for by this design include, but are not limited to,
friction, engine torque, machine center of gravity, and guide
handle position. As a result, dynamic balancing and consequent
force/torque reduction were found to result when the machine's
center of gravity was shifted substantially relative to a typical
machine's center of gravity. Dynamic balancing can be achieved most
practically by reversing the orientation of the engine relative to
the guide handle assembly when compared to traditional walk behind
rotary trowels and shifting the engine as far as practical to the
right. This shifting has been found to reduce the operational
forces and torque the operator must endure by at least 50% when
compared to traditional machines.
Inventors: |
Lutz, Todd J.; (Oconomowoc,
WI) ; Kruepke, Gregory; (Waukesha, WI) ;
Dauffenbach, Darrin W.; (Pewaukee, WI) ; Goldberg,
Richard D.; (Hartford, WI) |
Correspondence
Address: |
Timothy E. Newholm
BOYLE, FREDRICKSON, NEWHOLM, STEIN & GRATZ, S.C.
250 Plaza, Suite 1030
250 East Wisconsin Avenue
Milwaukee
WI
53202
US
|
Assignee: |
Wacker Corporation
|
Family ID: |
34435587 |
Appl. No.: |
10/704105 |
Filed: |
November 7, 2003 |
Current U.S.
Class: |
404/112 |
Current CPC
Class: |
E04F 21/248
20130101 |
Class at
Publication: |
404/112 |
International
Class: |
E01C 019/22 |
Claims
1. A concrete finishing trowel comprising: (A) a frame; (B) a motor
that is mounted on said frame and that has a rotatable output; (C)
an operator controlled guide handle that that extends rearwardly
from the frame along a line that at least generally laterally
bisects said frame; and; (D) a rotor that includes a plurality of
blades which are rotatable about a downwardly extending rotational
axis that is located on said line and that is at least
approximately centered on the frame, wherein said trowel is
dynamically balanced such that forces transmitted to the handle
upon rotation of the blades in contact with a surface to be
finished are substantially reduced when compared to a
non-dynamically balanced trowel, wherein said trowel has a center
of gravity that is offset longitudinally behind the rotational axis
of the rotor and laterally to the right of said line when viewed
from behind the trowel.
2. (canceled)
3. The trowel as recited in claim 1, wherein the trowel is a 36"
trowel, and the trowel's center of gravity is located between 0.00"
and 2.00" to right of the rotational axis of the rotor.
4. The trowel as recited in claim 3, and wherein the trowel's
center of gravity is located between 2.00" and 4.50" behind the
rotational axis of the rotor.
5. The trowel as recited in claim 4, wherein the trowel's center of
gravity is located about 0.75" to the right and about 3.875" behind
the rotational axis of the rotor.
6. The trowel as recited in claim 1, wherein the trowel is a 48"
trowel, and wherein the trowel's center of gravity is located
between 0.00" and 1.50" to the right of the rotational axis of the
rotor.
7. The trowel as recited in claim 6, wherein the trowel's center of
gravity is located between 2.00" and 4.50" behind the rotational
axis of the rotor.
8. The trowel as recited in claim 7, wherein the trowel's center of
gravity is located about 0.375" to the right and about 3.750"
behind the rotational axis of the rotor.
9. A concrete finishing trowel comprising: (A) a frame: (B) a motor
that is mounted on said frame and that has a rotatable output; (C)
an operator controlled guide handle that that extends rearwardly
from the frame along a line that at least generally laterally
bisects said frame; and (D) a rotor that includes a plurality of
blades which are rotatable about a downwardly extending rotational
axis that is located on said line and that is at least
approximately centered on the frame, wherein said trowel is
dynamically balanced such that forces transmitted to the handle
upon rotation of the blades in contact with a surface to be
finished are substantially reduced when compared to a
non-dynamically balanced trowel, wherein said trowel has a center
of gravity that is offset longitudinally behind the rotational axis
of the rotor and laterally to the right of said line when viewed
from behind the trowel, and wherein said motor has an output shaft
facing to the right of said trowel when viewed from behind the
trowel and a muffler facing forwardly of said trowel.
10. The trowel as recited in claim 1, wherein the longitudinal and
lateral offsets are selected in dependence on one another.
11. The trowel as recited in claim 9, wherein the longitudinal and
lateral offsets are selected based at least in part on at least one
of the following equations: 7 F 23 = dF w - a 2 b ( F w - F 45 ) -
b F 45 ( h - ea b ) where: F.sub.23=the combined longitudinal
forces imposed on the guide handle; d=the longitudinal offset;
F.sub.w=the gravitational force through the center of gravity of
the trowel; a=the length of a horizontal line connecting the
rotational axis of the rotor to the centroid of the forces acting
on one of the trowel blades, "a" being assumed to be the same for
each trowel blade; b=the longitudinal distance between the
rotational axis of the trowel and the guide handle; F.sub.45=the
combined vertical forces imposed on the guide handle; h=the height
of the guide handle; e=1/2 the lateral length of the guide handle;
.mu.=the dynamic coefficient of friction of the finished surface;
and 8 F 45 = F w ( b 2 hc - ceab - h 2 a 2 b + hea 2 + ehb d - eh a
2 + ab 2 d - a 3 b ) ( - h 2 a 2 + hea 2 - eh a 2 + ehb 2 - a 3 b +
ab 3 ) where: c=the lateral offset.
12. The trowel as recited in claim 1, wherein the lateral and
longitudinal offsets are determined taking guide handle length and
position and typical torque-generated forces into account.
13. The trowel as recited in claim 12, wherein the lateral and
longitudinal offsets are determined taking finished surface
coefficient of friction into account.
14. The trowel as recited in claim 1, wherein the longitudinal
offset is determined taking the following equation into account. 9
d = a 2 b Where: d=the longitudinal offset; a=the length of a
horizontal line connecting the rotational axis of the rotor to the
centroid of the forces acting on one of the trowel blades, "a"
being assumed to be the same for each trowel blade; and b=the
longitudinal distance between the rotational axis of the trowel and
the guide handle.
15. The trowel as recited in claim 1, wherein the lateral offset is
determined taking the following equation into account. 10 c = h a b
where: c=the lateral offset; h=the height of the guide handle;
a=the length of a horizontal line connecting the rotational axis of
the rotor to the centroid of the forces acting on one of the trowel
blades, "a" being assumed to be the same for each trowel blade;
.mu.=the dynamic coefficient of friction of the finished surface;
and b=the longitudinal distance between the rotational axis of the
trowel and the guide handle.
16. The trowel as recited in claim 1, wherein, during a concrete
finishing operation, the trowel is configured to impose an average
rearward force on the guide handle of no more than 50 lbs.
17. The trowel as recited in claim 16, wherein, during a concrete
finishing operation, the trowel is configured to impose an average
rearward force on the guide handle of no more than 30 lbs.
18. A concrete finishing trowel comprising: (A) a frame; (B) a
motor that is mounted on said frame; (C) an-operator controlled
guide handle that that extends rearwardly from the frame along a
line that at least generally laterally bisects said frame; and; (D)
a rotor that includes a plurality of blades which are rotatable
about a downwardly extending rotational axis that is located on
said line and that is at least approximately centered on the frame,
wherein said trowel has a center of gravity that is offset
longitudinally behind the rotational axis of the rotor and
laterally to the right of said line.
19. A concrete finishing trowel comprising: (A) a frame: (B) a
motor that is mounted on said frame; (C) an-operator controlled
guide handle that that extends rearwardly from the frame along a
line that at least generally laterally bisects said frame; and (D)
a rotor that includes a plurality of blades which are rotatable
about a downwardly extending rotational axis that is located on
said line is and that is at least approximately centered on the
frame, wherein said trowel has a center of gravity that is offset
longitudinally behind and laterally to the right of the rotational
axis of the rotor when viewed from behind the trowel, wherein said
motor has an output shaft facing to the right of said trowel when
viewed from behind the trowel and a muffler facing forwardly of
said trowel.
20. A concrete finishing trowel comprising: (A) a frame; (B) a
motor that is mounted on said frame; (C) an operator controlled
guide handle that that extends rearwardly from the frame, and; (D)
a rotor that includes a plurality of blades which are rotatable
about a downwardly extending rotational axis that is at least
approximately centered on the frame, wherein said motor has an
output shaft facing to the right of said trowel when viewed from
behind the trowel and a muffler facing forwardly of said
trowel.
21-32. (canceled)
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The invention relates to concrete finishing trowels and,
more particularly, relates to a walk-behind rotary concrete
finishing trowel which is dynamically balanced to reduce operator
effort. The invention additionally relates to a method of operating
such a trowel.
[0003] 2. Discussion of the Related Art
[0004] Walk behind trowels are generally known for the finishing of
concrete surfaces. A walk behind trowel generally includes a rotor
formed from a plurality of trowel blades that rest on the ground.
The rotor is driven by a motor mounted on a frame or "cage" that
overlies the rotor. The trowel is controlled by an operator via a
handle extending several feet from the cage. The rotating trowel
blades provide a very effective machine for finishing mid-size and
large concrete slabs. However, walk behind trowels have some
drawbacks.
[0005] For instance, the rotating blades impose substantial
forces/torque on the cage that must be counteracted by the operator
through the handle. Specifically, blade rotation imposes a torque
on the cage and handle that tends to drive the handle to rotate
counterclockwise or to the operator's right. In addition, blade
rotation tends to push the entire machine linearly, principally
backwards, requiring the operator to push forward on the handle to
counteract those forces. The combined torque/forces endured by the
operator are substantial and tend to increase with the dynamic
coefficient of friction encountered by the rotating blades which,
in turn, varies with the "wetness" of curing concrete.
Counteracting these forces can be extremely fatiguing, particularly
considering the fact that the machine is typically operated for
several hours at a time.
[0006] The inventors investigated techniques for reducing the
reaction forces/torque that must be endured by the operator. They
theorized that these forces would be reduced if the trowel were
better statically balanced than is now typically the case with walk
behind trowels, in which the center of gravity is located slightly
behind and to the left of the rotor's axis of rotation. The
inventors therefore theorized that shifting the trowel's center of
gravity forwardly would reduce reaction forces. However, they found
that this shifting actually led to an increase in reaction forces
generated during trowel operation.
[0007] The need therefore has arisen to provide a walk behind
rotary trowel that requires substantially less operator effort to
steer and control than conventional walk behind trowels.
[0008] The need additionally has arisen to reduce the operator
effort required to steer and control a walk behind rotary
trowel.
SUMMARY OF THE INVENTION
[0009] Pursuant to the invention, a walk behind rotary trowel is
configured to be better "dynamically balanced" so as to minimize
the forces/torque that the operator must endure to control and
guide the trowel. The design takes into account both static and
dynamic operation and attributes of the trowel, and "balances"
these attributes with the operational characteristics of concrete
finishing. Characteristics that are accounted for by this design
include, but are not limited to, friction, engine torque, machine
center of gravity, and guide handle position. As a result, dynamic
balancing and consequent force/torque reduction were found to
result when the machine's center of gravity was shifted
substantially relative to a typical machine's center of gravity.
This effect can be achieved most practically by reversing the
orientation of the engine relative to the guide handle assembly
when compared to traditional walk behind rotary trowels and
shifting the engine as far as practical to the right. This shifting
has been found to reduce the operational forces and torque the
operator must endure by at least 50% when compared to traditional
machines. Operator fatigue therefore is substantially reduced.
[0010] These and other advantages and features of the invention
will become apparent to those skilled in the art from the detailed
description and the accompanying drawings. It should be understood,
however, that the detailed description and accompanying drawings,
while indicating preferred embodiments of the present invention,
are given by way of illustration and not of limitation. Many
changes and modifications may be made within the scope of the
present invention without departing from the spirit thereof, and
the invention includes all such modifications.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] A preferred exemplary embodiment of the invention is
illustrated in the accompanying drawings in which like reference
numerals represent like parts throughout, and in which:
[0012] FIG. 1 is a perspective view of a walk-behind rotary trowel
constructed in accordance with a preferred embodiment of the
present invention;
[0013] FIG. 2 is a side elevation view the trowel of FIG. 1;
[0014] FIG. 3 is a front elevation view of the trowel of FIGS. 1
and 2;
[0015] FIG. 4 is a series of graphs charting force v. RPM for a
variety of operating conditions; and
[0016] FIGS. 5A-5C are a series of force diagrams that
schematically illustrate the forces generated upon operation of a
walk behind trowel.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0017] 1. Construction of Trowel
[0018] A walk behind trowel 10 constructed in accordance with a
preferred embodiment of the invention is illustrated in FIGS. 1-3.
In general, the walk behind trowel 10 includes a rotor 12, a frame
or "cage" 14 that overlies and is supported on the rotor 12, an
engine 16 that is supported on the cage 14, a drive train 18
operatively coupling the engine 16 to the rotor 12, and a handle 20
for controlling and steering the trowel 10. Referring to FIG. 2,
the rotor 12 includes a plurality of trowel blades 22 extending
radially from a hub 24 which, in turn, is driven by a vertical
shaft 26.
[0019] The motor 16 comprises an internal combustion engine mounted
on the cage 14 above the rotor 12. Referring again to FIGS. 1-13,
the engine 16 is of the type commonly used on walk behind trowels.
It therefore includes a crankcase 30, a fuel tank 32, an air supply
system 34, a muffler 36, a pull-chord type starter 38, an output
shaft (not shown), etc. The drive train 18 may be any structure
configured to transfer drive torque from the engine output shaft to
the rotor input shaft 26. In the illustrated embodiment, it
comprises a centrifugal clutch (not shown) coupled to the motor
output shaft and a gearbox 40 that transfers torque from the clutch
to the rotor input shaft 26. The gearbox is coupled to the clutch
by a belt drive assembly 42, shown schematically in FIG. 1. The
preferred gearbox 40 is a worm gearbox of the type commonly used on
walk behind trowels.
[0020] The handle assembly 12 includes a post 44 and a guide handle
46. The post 44 has a lower end 48 attached to the gearbox 40 and
an upper end 50 disposed several feet above and behind the lower
end 48. The guide handle 46 is mounted on the upper end 50 of the
post 44. A blade pitch adjustment knob 52 is mounted on the upper
end 50 of the post 44. Other controls, such as throttle control, a
kill switch, etc., may be mounted on the post 44 and/or the guide
handle 46.
[0021] The cage 14 is formed from a plurality of vertically spaced
concentric rings 54 located beneath a deck 56 and interconnected by
a number of angled arms 58, each of which extends downwardly from
the bottom of the deck 56 to the bottommost rings 54. The rings 54
may be made from tubes, barstock, or any other structure that is
suitably rigid and strong to support the trowel 10 and protect the
rotor 12. In order to distribute weight in a desired manner, one or
more of the rings 54 may be segmented, with one or more arcuate
segment(s) being made of relatively light tubestock, other
segment(s) being made of heavier barstock, and/or other segment(s)
being eliminated entirely. One or more of the arm(s) 58 could be
similarly segmented. Weights could also be mounted on the cage 14
at strategic locations to achieve additional strategic weight
distribution.
[0022] 2. Center of Gravity Offset
[0023] Still referring to FIGS. 1-3, and in accordance with the
invention, the trowel's center of gravity "C/G" is offset laterally
and longitudinally relative to the rotor's rotation axis "A."
Specifically, the center of gravity is spaced rearwardly and to the
right of the rotational axis A. The considerations behind this
positioning and the optimal positions are discussed in more detail
in Section 3 below. In the illustrated embodiment, practical
dynamical balancing is best achieved through two effects. First,
the engine 16 is rotated 180.degree. relative to the guide handle
20 when compared to a conventional machine. Hence, the fuel tank 32
faces rearwardly, or towards the operator, and the air supply
system 34 and muffler 36 face forwardly, away from the operator. In
addition, the torque transfer system 18 is positioned to the
operator's right as opposed to his or her left, and the pull chord
38 is positioned on the operator's left as opposed to his or her
right. The engine 16 therefore can be considered "forward facing"
as opposed to "rearward facing." As a result, the engine's center
of gravity C/G is disposed to the right of trowel's geometric
center. The gearbox 40 is also rotated 180.degree. to accommodate
the engine's reorientation. The combined effect of these
reorientations is a significant shift of the machine's center of
gravity C/G to the right when compared to prior machines. It also
moves the center of gravity C/G to a location further behind the
rotor's rotational axis A.
[0024] In the illustrated embodiment of a 48" trowel, i.e., one
whose blade circumference is a 48" diameter circle, optimal results
given the practical limitations of the machine design, such as
guide handle length, engine mass, limitations on engine to gearbox
spacing, etc., resulted when the engine 16 was shifted so as to
shift or relocate the center of gravity C/G to a location 3.75
inches behind and 0.375 inches to the right of the trowel axis A.
The resultant longitudinal and lateral offsets, "d" and "c", are
illustrated in FIGS. 2 and 3, respectively. Of course, some of the
beneficial balancing effects would result with smaller offsets,
particularly smaller lateral (X) offsets, such as 0.125. Optimum
offset calculations and offset interdependence are discussed in
section 3 below.
[0025] This relocation has been found to nearly eliminate the
linear forces acting on the guide handle 46, requiring that the
operator only need to counteract the rotational torque imposed on
the handle and the linear forces resulting from that torque. This
effect is illustrated in the series of graphs of FIG. 5, which
compare the forces and endured by an operator of a prior art 48"
trowel to those imposed by a trowel constructed as described above.
The forces were measured with standard blades operating on a steel
sheet. A comparison of curves 60 to 64 confirm that, depending on
engine RPM, total forces endured are reduced from about 65-75 lbs,
to 20-30 lbs. A comparison of curves 62 and 66 reveals that linear
forces, i.e., those resulting from factors other than blade torque
and compensated for by offsetting the machine's center of gravity
as described above, are reduced from about 40-45 lbs to less than
10 lbs.
[0026] An ancillary benefit of this engine reorientation is that it
increases operator comfort because the heat and fumes from the
exhaust are now directed away from the operator rather than towards
the operator.
[0027] 3. Center of Gravity Offset Determination
[0028] The optimal lateral and longitudinal center of gravity
offsets "c" and "d" relative to the rotor's rotational axis A,
i.e., the optimal center of gravity position for a given trowel
design, could be determined purely empirically by trial and error.
They could also be determined mathematically by taking practical
considerations into account, such as machine geometry and changes
in coefficient of dynamic friction experienced by the trowel during
the curing concrete process, etc. These calculations will now be
explained with reference to FIGS. 5A-5C, which schematically
illustrate the forces generated during operation of the walk behind
trowel.
[0029] Dynamically balancing the trowel requires that as many
forces acting on the handle as possible be eliminated. Referring
first to FIG. 5A, which is a force diagram in the horizontal (XY)
plane, the lines 70 designate the blades, it being assumed that
each blade has the same effective length "a," as measured from the
rotor rotational axis A to the centroid of the forces acting on the
trowel blade. The line 72 designates the handle in the lateral (X)
plane and has effective lengths "e" on either side of the center
post 44 (FIGS. 1-3), i.e., the guide handle and has a lateral
length of 2 e. The handle 12 has an effective longitudinal length
"b," as measured from the rotational axis A of the rotor to the
grips on the guide handle as schematically represented by the line
74. In operation, the four blades are subjected to
friction-generated horizontal forces F.sub.Af, F.sub.Bf, F.sub.Cf,
and F.sub.Df, respectively, which result in corresponding moment
arms aF.sub.Af, aF.sub.Bf, aF.sub.Cf, and aF.sub.Df about the rotor
axis A. The handle 12 is subjected to longitudinal (Y) horizontal
forces F.sub.H2 and F.sub.H3 and a lateral (X) force F.sub.H1.
[0030] The forces acting on the handle in the X direction can
balanced or set to zero using the equation:
F.sub.H1+F.sub.Af=F.sub.Bf Equation 1
[0031] The forces acting on the handle in the Y direction can
balanced or set to zero using the equation:
F.sub.Cf=F.sub.Df+F.sub.H2+F.sub.H3 Equation 2
[0032] The moment in the XY plane can be balanced or set to zero
using the equation:
a(F.sub.Af+F.sub.Bf+F.sub.Cf+F.sub.Df)=bF.sub.H1+eF.sub.H2-eF.sub.H3
Equation 3
[0033] The same procedure can be used to represent the balancing of
forces in the remaining planes. Hence, referring to FIG. 5B, which
represents the trowel in the XZ plane, the vertical (Z) forces
acting on the handle can balanced or set to zero using the
equation:
F.sub.w=F.sub.AZ+F.sub.BZ+F.sub.CZ+F.sub.DZ+F.sub.H4+F.sub.H5
Equation 4
[0034] Where, in addition to the forces defined above:
[0035] F.sub.AZ, F.sub.BZ, F.sub.CZ, and F.sub.DZ=the vertical
forces acting on the blades;
[0036] F.sub.H4 and F.sub.H5=the vertical forces acting on the ends
of the guide handle;
[0037] F.sub.w=the gravitational force acting through the machine's
center of gravity; and
[0038] c=the lateral (X) offset between the machine's center of
gravity C/G and the center of the machine, which coincides with the
rotor axis of rotation A.
[0039] The moment in the XZ plane can be balanced or set to zero
using the equation:
aF.sub.Dz+hF.sub.H1+eF.sub.H5-eF.sub.H4-aF.sub.Cz-cF.sub.w=0
Equation 5
[0040] Where: h=height of the guide handle (see line 76 in FIG.
5B).
[0041] Referring to FIG. 5C, which represents the trowel in the YZ
plane, the moment in the YZ plane can be balanced or set to zero
using the equation:
aF.sub.AZ+dF.sub.w=aF.sub.Bz+bF.sub.A4+bF.sub.A5+hF.sub.H2+hF.sub.H3
Equation 6
[0042] Where: d=the longitudinal (Y) offset between the machine's
center of gravity C/G and the center of the machine, which
coincides with the rotor axis of rotation A.
[0043] Using the above parameters, the side-to-side center of
gravity, c, as a function of forces on the handle, the trowel
dimensions, and the coefficient of friction, .mu., of the surface
to be finished, can be expressed as: 1 hF H1 + e ( F H5 - F H4 ) -
[ bF H1 + e ( F H2 - F H3 ) 2 ( F w - F H4 - F H5 ) ] ( F H2 + F H3
) F w = c Equation 7
[0044] The force F.sub.H1 results for torque imposed by blade
rotation and cannot be eliminated by adjusting the trowel's center
of gravity. However, by simplifying equation 7 to set the remaining
forces F.sub.H2, F.sub.H3, F.sub.H4, and F.sub.H5 to zero, the
lateral offset, c, required to eliminate those forces can be
determined by the equation: 2 c = h a b Equation 8
[0045] Similarly, the front-to-rear center of gravity, d, as a
function of forces imposed on the handle, the trowel dimensions,
and the finished surface coefficient of friction, .mu., can be
expressed as: 3 d = bF H1 2 + eF H1 ( F H2 - F H3 ) 2 ( F w - F H4
- F H5 ) + b ( F H4 + F H5 ) + h ( F H2 + F H3 ) F w Equation 9
[0046] By simplifying equation 9 to set the forces F.sub.H2,
F.sub.H3, F.sub.H4, and F.sub.H5 to zero, Equation 9 can be solved
for d using the equation: 4 d = a 2 b Equation 10
[0047] Hence, a machine configured to have a center of gravity C/G
that is laterally and longitudinally offset from the center of the
machine (as determined by the rotor's axis of rotation A) by values
c and d as determined using equations 8 and 10 would theoretically
impose no non-torque induced forces on the handle during trowel
operation.
[0048] The theoretical values of c and d are not practical for most
existing walk-behind trowel configurations and might not even be
possible for some trowels. For instance, the theoretical best
lateral offset c might be spaced so far from the rotor rotational
axis A that the engine would have to be cantilevered off the side
of the machine.
[0049] As such, it is necessary as a practical matter to determine
the effects that c and d have on each other over a range of offsets
and to select practical values of c and d that best achieve the
desired goal of dynamic balancing. This can be done using the
followings steps:
[0050] First, to simplify the calculations by discounting the least
problematic forces to the extent that they are minimal and/or
relatively unlikely to occur, it can be assumed that no twisting
forces are imposed on the guide handle 46 (i.e., F.sub.H4=F.sub.H5)
and that F.sub.H3=0 due to the fact that the operator typically
pushes on the handle with only the left hand to be counteract the
torque imposed by the clockwise rotating blades. The combined force
F.sub.23 (resulting from the combination of the longitudinal forces
F.sub.H2 and F.sub.H3) can be determined for each of a number of
practical longitudinal offsets d using the following equation: 5 F
23 = dF w - a 2 b ( F w - F 45 ) - b F 45 ( h - ea b ) Equation
11
[0051] Second, the combined force F.sub.45 (resulting from the
combination of the vertical forces F.sub.H4 and F.sub.H5) can be
determined for each of a number of practical longitudinal offsets d
and practical lateral offsets c using the following equation: 6 F
45 = F w ( b 2 hc - ceab - h 2 a 2 b + hea 2 + ehb d - eh a 2 + ab
2 d - a 3 b ) ( - h 2 a 2 + hea 2 - eh a 2 + ehb 2 - a 3 b + ab 3 )
Equation 12
[0052] A table can then be generated that permits the designer to
select the offsets c and d that strike the best balance between
F.sub.23 and F.sub.45. Of course, the designer may choose to place
priority on one of these values, for instance by selecting an
offset that reduces F.sub.45 as much as practical while sacrificing
some reduction in F.sub.23.
[0053] The effects of this analysis and its practical
implementation can be appreciated from Table 1, which relays
traditional typical (prior art) offsets, theoretical offsets, and
practical offsets as selected using the procedure described
immediately above for both a 36" trowel and a 48" trowel, where
positive values indicate locations behind or to the right of the
rotor axis A and negative values indicate locations ahead or to
left of the rotor axis A. Note that the terms "36 inch trowel" and
"48 inch trowel" are accepted terms of art designating standard
trowel sizes rather than designating any particular precise trowel
dimension. Note also that a few manufacturers refer to what is more
commonly known as a "48 inch trowel" as a "46 inch trowel."
1TABLE 1 Typical Offsets 36" Trowel 48" Trowel Standard x offset
-0.375" -0.125 Standard y offset 3.25" 2.50" Theoretical x offset
3.46" 3.88" Theoretical y offset 1.59" 2.38" Typical practical x
offset 0.75" 0.375" Typical practical y offset 3.875" 3.75"
[0054] 4. Operation of Trowel
[0055] During normal operation of the trowel 10, torque is
transferred from the engine's output shaft, to the clutch, the
drive train, the gearbox 40, and the rotor.
[0056] The blades 22 are thereupon driven to rotate and contact
with the surface to be finished, smoothing the concrete. The
frictional resistance imposed by the concrete varies, e.g., with
the rotor rotation or velocity, the types of blades or pans used to
finish the surface and the orientation of the blades or pan
relative to the surface, and the coefficient of friction of the
surface. The operator guides the machine 10 along the surface
during this operation using the guide handle. In prior walk behind
trowels, this operation would be resisted by substantial forces
totaling 60-75 lbs. However, because the trowel 10 is dynamically
balanced as described above, the total forces endured by the
operator to 20-30 lbs., a reduction of well over 50%.As indicated
above, many changes and modifications may be made to the present
invention without departing from the spirit thereof. The scope of
some of these changes is discussed above. The scope of others will
become apparent from the appended claims.
* * * * *